The convex-analysis tag has no wiki summary.

**11**

votes

**0**answers

153 views

### How large are the smallest-area projections of a high-dimensional convex body?

Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that
$$\intop_B x \, dx = 0$$
$$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$
a ...

**6**

votes

**2**answers

128 views

### On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: http://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse
On the convexity of element-wise norm 1 of the ...

**5**

votes

**2**answers

243 views

### Efficient computation of “discrete infimal convolution”

This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...

**3**

votes

**3**answers

180 views

### infinite dimensional polyhedra

I have a reference request which I hope some reader here can help me with.
I have encountered a set that has all the properties that one would expect from a polyhedral set (in the sense of finite ...

**3**

votes

**3**answers

318 views

### When is a sequentially closed cone, closed?

The following question I also posed here, but still got no answer. Let $X$ be a locally convex, Hausdorff topological vector space and $C\subseteq X$ a convex cone, which is sequentially closed. What ...

**3**

votes

**1**answer

237 views

### Possible to find a set of log-concave functions with log-concave sums?

While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...

**3**

votes

**1**answer

155 views

### Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.
\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ ...

**3**

votes

**1**answer

348 views

### Quantitative Version of Jensen's Inequality?

Hi,
I've been looking at situations where Jensen's inequality is almost tight, and found myself proving a lemma that I'm nearly certain exists somewhere in the literature.
The specifics are as ...

**3**

votes

**0**answers

143 views

### Ways to establish equality of measures on locally compact spaces

Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality
$$
...

**2**

votes

**2**answers

346 views

### Does the minima of a sequence of convex convergent functions converge?

Suppose $f_1,f_2,\ldots $ is a sequence of convex functions that converges to a continuous convex $f$. Let $x_1^*,x_2^*$ be their respective (not necessarily unique) minima, and let y be a minima of ...

**2**

votes

**5**answers

187 views

### Distance between two sets

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem.
$$ \min ...

**2**

votes

**2**answers

84 views

### Projection onto rotated box

Does anyone know if there is an efficient way to find the projection of an arbitrary point $z$ onto a rotated box, i.e. onto the set $\Omega=\{x \mid a \leq Ux \leq b\}$ where $U$ is a unitary matrix?
...

**2**

votes

**1**answer

581 views

### Projection exists => Uniformly convex?

Hello,
I know that: Let X be a uniformly convex Banach-Space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x ...

**2**

votes

**2**answers

125 views

### Perimeter of a 'trapped' convex set

Consider the following setup: three bounded, 'nice' convex sets $A \subseteq B \subsetneq C \subset \mathbb{R}^2$, and three points $x,y,z\in \partial A\cap \partial B\cap \partial C$ (see edit ...

**2**

votes

**1**answer

166 views

### How to examine the convexity of a complex function numerically?

I have a function which does not have a closed form . Large numerical effort should be done to evaluate the function for even a single point. How can I examine the convexity of my function over the ...

**2**

votes

**1**answer

378 views

### Integral inequality for convex function

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true:
\begin{equation}
\frac{1}{b-a} \int_a^b ...

**2**

votes

**2**answers

223 views

### A certain type of Quadratic Constrained Quadratic Programming Problem (QCQP)

Let $P_1$, $P_2$ be two hermitian matrices. Can anyone comment about the following (QCQP)
\begin{equation}
\min_{z}~z^{H}z \\\
~~subject~to ~z^{H}P_1z+1\leq 0,~z^{H}P_2z+1\leq 0
\end{equation}
I am ...

**2**

votes

**1**answer

967 views

### Sufficient conditions for gradient descent convergence

I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps ...

**2**

votes

**1**answer

311 views

### existence of a minimum for a convex functional on a non-reflexive space

Let $X$ be a Banach space; $K\subset X$ nonempty, closed and convex; and $f:K\to \mathbb R$ lower semicontinuous, convex functional. Let also $f$ be coercive, i.e., $f(x)\to +\infty$ as $\|x\|\to ...

**2**

votes

**1**answer

65 views

### Known Results on Convexity of Numerical Range

Let $A_1,A_2,\dots,A_M$ be given $N\times N$ hermitian matrices. The numerical range is defined as the set
\begin{align}
\mathbb{S}=\{(u^HA_1u,\dots,u^HA_Mu)\in \mathbb{R}^M\mid u^Hu=1\}
\end{align}
...

**2**

votes

**1**answer

275 views

### Proving that a specific function is quasiconvex

Hello all,
Assume we have a sequence of quasiconcave functions (in $X$) denoted by $f_{i,j}(X)$ for $i,j = 1,\ldots,n$. Denote by $F(X)$ the $n\times n$ matrix whose $(i,j)$ entry is the function ...

**2**

votes

**1**answer

109 views

### The epigraph of a semi-convex function has positive reach

I've been trying to prove the following theorem for several hours with no result so far.
Claim. Let $f:\mathbb{R} \to \mathbb{R}$ be a semi-convex function, i.e. there exists a constant $C > 0$ ...

**2**

votes

**0**answers

83 views

### How to prove convexity for a complex integral including the variable in both limits and integrand?

During my research [on inventory management policies, i.e., something really applied ;-) ] I stumbled on integrals of the following type and I'm curious under which circumstances there are convex.
...

**1**

vote

**1**answer

158 views

### Are all points x of the boundary of a convex set C of a Hilbert space H projections onto C of a point different than x?

It appears that this is not true if H is of infinite dimension. My question is therefore the following: does anyone have a counter-example? Is there a caracterisation for the points of the boundary ...

**1**

vote

**1**answer

187 views

### positive semigroups and convex operator

Let $\{Z(t)\}_{t\geq 0}$ be a strongly continuous positive semigroup on a Banach lattice V. (endowed with ordering $\leq$). $\phi:V\rightarrow V$ is a convex operator. I want to prove that ...

**1**

vote

**1**answer

218 views

### When does the finite union of convex sets have a hole in it?

Let $f_1, \dots, f_j$ be convex functions from $\mathbb{R}^n \to \mathbb{R}$. I am trying to develop a test that decides whether or not the set $\{x | f_1(x) \le k_1\} \cup \dots \cup \{x | f_n(x) ...

**1**

vote

**2**answers

130 views

### determining a convex set by mixed volumes

For a convex set $K \subset \mathbb{R}^2$ let $\phi_K:$ convexsets in $\mathbb{R}^2 \rightarrow [0,\infty), A \mapsto MV(A,K)$. Where by $MV(A,K)$ I mean the mixed volume of $A$ and $K$ in the ...

**1**

vote

**1**answer

97 views

### Relative interior and dense subsets

(This is a cross-post from here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ by ...

**1**

vote

**2**answers

364 views

### Hessian of function of covariance matrices

Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$,
$$
\mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...

**1**

vote

**1**answer

228 views

### How does the complex convex set look like?

The usual convex set is the real linear convex set, if we change the real linear map into complex linear map, we can get the complex convex set. A system way to do this is in the several complex ...

**1**

vote

**2**answers

142 views

### What is the dual of an semidefinitely representable (SDR) cone?

The Question
Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product.
Let $\mathcal K\subset V$ be a cone defined as
$$
\mathcal K=\Big\{x\in ...

**1**

vote

**1**answer

228 views

### Question regard checking convexity by “restriction to any line that intersects the function domain”

Hello all,
I have a question (probably stupid one) about the fact that " A function is convex if and only if it is convex when restricted to any line that intersects its domain".
In Stephen Boyd and ...

**1**

vote

**0**answers

45 views

### Directional derivates and unique subgradients

I have a question about the fine structure of convex functions. Convex functions behave very regular in the interior of their domain of definition (e.g. they are locally Lipschitz continuous there) ...

**1**

vote

**0**answers

57 views

### monotone parabolic systems, convex variational structure and Legendre transform

The context:
for my research I am currently looking at parabolic systems of the type
$$
\left\{
\begin{array}{ll}
\partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\
u=0 & ...

**1**

vote

**0**answers

91 views

### Lipschitz continuity of solution set mapping of a parametric convex optimization problem

I have a parametric convex optimization problem:
\begin{array}{cl}
\underset{x}{\text{minimize}} & f\left(x,z\right)\\
\text{subject to} & g\left(x\right)\leq0
\end{array}
where $x$ is the ...

**1**

vote

**1**answer

96 views

### Does John's Ellipsoid preserve subset ordering? [duplicate]

Let $K \subset \mathbb{R}^d$ be a convex body, symmetric about the origin and with nonempty interior. Then John's theorem asserts that there exists a unique ellipsoid $E$ of minimal volume such that ...

**1**

vote

**0**answers

111 views

### An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking.
These subgradients are (assume $x \in$ ...

**1**

vote

**0**answers

86 views

### Linearization of cones

Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?

**1**

vote

**0**answers

83 views

### Concavity of a ratio of Kullback-Leibler divergences

For $0\leq r\leq 1$ and $0 < p < 1$, define the Kullback-Leibler divergence between the Bernoulli(r) and Bernoulli(p) distributions as: $D(r||p) := r\log\frac{r}{p} + ...

**1**

vote

**0**answers

109 views

### Optimize a convex hull on a 2D histogram so the selected points match a target shape

I have an image (can be 2D or 3D), and compute a 2D histogram of the image (for example, the pixel intensity and gradient along certain direction). There is a known target region $R^*$ in the image. I ...

**1**

vote

**0**answers

86 views

### Subgradient of Minimum Eigenvalue

Consider three $N \times N$ hermitian matrices $A_0,A_1,A_2$. Consider the function
\begin{align}
f(t_1,t_2)=\lambda_{min}(A_0+t_1A_1+t_2A_2)
\end{align}
where $\lambda_{min}$ denotes the minimum ...

**1**

vote

**0**answers

340 views

### Definition and Convergence of Iteratively Reweighted Least Squares

I've been using iteratively reweighted least squares (IRLS) to minimize functions of the following form,
$J(m) = \sum_{i=1}^{N} \rho \left(\left| x_i - m \right|\right)$
where $N$ is the number of ...

**0**

votes

**1**answer

160 views

### A certain type of quadratic problem.

I am interested in solving the following equality constrained quadratic (?) problem.
\begin{align}
\min_{u^{H}u=1}~(u^{H}A_1u) \\\
s.t.~ u^{H}A_2u=0
\end{align}
$A_1$ and $A_2$ are $N\times N$ ...

**0**

votes

**1**answer

128 views

### Exponential Convexity

$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous
and
$$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$
for all $n\in\mathbb{N}$ and all ...

**0**

votes

**2**answers

87 views

### Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints

Let $C_1$,$C_2$,...$C_N$ be $M \times M$ indefinite hermitian matrices. What can we say about the following quadratic constriants
\begin{align}
w^{H}C_1w>0 \\\
w^{H}C_2w>0 \\\
...~~~~~~~~~~ \\\
...

**0**

votes

**1**answer

121 views

### Minimal point of the intersection of convex sets.

I am trying to find out if there is any known result in convex optimization that implies the following statement:
"A minimal point of the intersection of $N > 2$ convex sets in $\mathbb{R}^2$ is ...

**0**

votes

**1**answer

98 views

### Is these two optimization problems share the same solution?

Hello all,
I am dealing with some SDP optimization, and I come across the following problem.
The optimization problem is given by
\begin{aligned}
&\operatorname*{min}_{t_1,\ldots,t_m,X}\ \sum ...

**0**

votes

**1**answer

619 views

### Is a jointly convex function of x and y convex as a function of x when y=z(x)?

Hi,
Suppose that $x \in R^m, y \in R^n, z(x) \in R^n$, and $f(x,y)$ is convex in $(x,y)$.
Is $f(x,z(x))$ a convex function in $x$ for arbitrary continuous functions $z(x)$?
Thanks!

**0**

votes

**1**answer

123 views

### Convex functions: bounding the difference

Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and $x_1, x_2, \ldots, x_n \in R^d$ such that
$$\sum_{i=1}^n (x_i - x') = x - x'.$$
Is it ...

**0**

votes

**0**answers

34 views

### additive support functions of a convex set

Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by
$$h_K(u) = \sup \{ \sum_{i=1}^d x_i ...